RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the...
Transcript of RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the...
![Page 1: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/1.jpg)
RECURRENCE:
RANDOM WALKS vs
QUANTUM WALKS
MARTES CUÁNTICO 05/05/2015
![Page 2: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/2.jpg)
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
![Page 3: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/3.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
![Page 4: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/4.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
particles independently change container at rate N ∆t
![Page 5: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/5.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
particles independently change container at rate N ∆t
![Page 6: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/6.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
N − k k
particles independently change container at rate N ∆t
![Page 7: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/7.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Even !!![N, 0]Any state has return probability R = 1
N − k k
particles independently change container at rate N ∆t
![Page 8: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/8.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Even !!![N, 0]Any state has return probability R = 1
N − k k
particles independently change container at rate N ∆t
Differences come from the expected return time
τ [N−k,k] =2N
(
N
k
) ∆t
![Page 9: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/9.jpg)
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Even !!![N, 0]Any state has return probability R = 1
N − k k
particles independently change container at rate N ∆t
Differences come from the expected return time
τ [N−k,k] =2N
(
N
k
) ∆t
millions of years ≈ 3×age of Universeτ[N,0] ≈ 40 000
secτ[N/2,N/2] ≈ 1.25× 10−11
For instance, if and sec N = 100 ∆t = 10−12
![Page 10: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/10.jpg)
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
![Page 11: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/11.jpg)
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
chaos non-equilibriummicro-thermodynamics
![Page 12: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/12.jpg)
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
chaos non-equilibriummicro-thermodynamics
![Page 13: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/13.jpg)
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
The Ehrenfest example shows that simple models (random, discrete) may help to uncover central aspects of recurrence avoiding unnecessary complications
chaos non-equilibriummicro-thermodynamics
![Page 14: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/14.jpg)
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
The Ehrenfest example shows that simple models (random, discrete) may help to uncover central aspects of recurrence avoiding unnecessary complications
RECURRENCEOLD RANDOM WALKS (RW)
chaos non-equilibriummicro-thermodynamics
![Page 15: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/15.jpg)
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
The Ehrenfest example shows that simple models (random, discrete) may help to uncover central aspects of recurrence avoiding unnecessary complications
RECURRENCEOLD RANDOM WALKS (RW)
NEW! QUANTUM WALKS (QW)
chaos non-equilibriummicro-thermodynamics
![Page 16: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/16.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
![Page 17: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/17.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
![Page 18: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/18.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
![Page 19: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/19.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
![Page 20: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/20.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
![Page 21: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/21.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
![Page 22: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/22.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
R = 1 R = 1 R ≈ 0.34
![Page 23: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/23.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
R = 1 R = 1 R ≈ 0.34CRITICAL
DIMENSIOND= 3
⇒UNBIASED
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
![Page 24: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/24.jpg)
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
R = 1 R = 1 R ≈ 0.34
τ = ∞ τ = ∞
CRITICAL DIMENSION
D= 3
⇒UNBIASED
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
![Page 25: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/25.jpg)
RW RECURRENCE
STATES: elements of a countable set Ωi ∈ Ω
![Page 26: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/26.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
![Page 27: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/27.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
![Page 28: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/28.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
![Page 29: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/29.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
![Page 30: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/30.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
![Page 31: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/31.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
![Page 32: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/32.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
![Page 33: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/33.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
![Page 34: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/34.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
Prob(in steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1j j
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
![Page 35: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/35.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1j jj
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
![Page 36: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/36.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
![Page 37: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/37.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
i
![Page 38: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/38.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
i
![Page 39: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/39.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
OVERCOUNTING!!!
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
![Page 40: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/40.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
![Page 41: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/41.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
![Page 42: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/42.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
![Page 43: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/43.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
![Page 44: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/44.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
![Page 45: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/45.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
![Page 46: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/46.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Ri = Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)RETURN
PROBABILITY
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
![Page 47: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/47.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Ri = Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
EXPECTED RETURN TIME
RETURNPROBABILITY
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
q(n)i∆t( )τi =
X
n≥1
n
i
i
SIMPLE LOOP
![Page 48: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/48.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Ri = Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
EXPECTED RETURN TIME
RETURNPROBABILITY
CONVENTION
∆t = 1[ [
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
q(n)i
τi =
X
n≥1
n
i
i
SIMPLE LOOP
![Page 49: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/49.jpg)
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)i
Ri =
EXPECTED RETURN TIME
RETURNPROBABILITY
CONVENTION
∆t = 1[ [
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
q(n)i
τi =
X
n≥1
n
i
i
SIMPLE LOOP
![Page 50: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/50.jpg)
RW RECURRENCE: GENERALITIES
![Page 51: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/51.jpg)
RW RECURRENCE: GENERALITIES
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
![Page 52: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/52.jpg)
RW RECURRENCE: GENERALITIES
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
![Page 53: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/53.jpg)
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
![Page 54: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/54.jpg)
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
![Page 55: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/55.jpg)
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
p(n)i
q(n)i≥
![Page 56: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/56.jpg)
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
p(n)i
q(n)i≥
![Page 57: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/57.jpg)
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
p(n)i
q(n)i≥
T R
![Page 58: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/58.jpg)
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
FINITE systems may have TRANSIENT states
p(n)i
q(n)i≥
T R
![Page 59: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/59.jpg)
Ω
SUBSET RECURRENCE
i
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
FINITE systems may have TRANSIENT states
p(n)i
q(n)i≥
T R
![Page 60: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/60.jpg)
Ω
SUBSET RECURRENCE
Si
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
FINITE systems may have TRANSIENT states
p(n)i
q(n)i≥
Prob(i → S) Prob(i → i)≥ = Ri
T R
![Page 61: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/61.jpg)
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities?
![Page 62: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/62.jpg)
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
![Page 63: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/63.jpg)
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
![Page 64: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/64.jpg)
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
![Page 65: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/65.jpg)
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
![Page 66: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/66.jpg)
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
![Page 67: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/67.jpg)
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
= bqi(1)
=
dbqidz
∣∣∣∣z=1
![Page 68: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/68.jpg)
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
= bqi(1)
=
dbqidz
∣∣∣∣z=1
= 1−1
bpi(1)
= limz→1
1
(1− z)bpi(z)
![Page 69: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/69.jpg)
RECURRENTi ∈ Ω Ri = 1⇔
POSITIVE RECURRENT ⇔ τi < ∞
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
= bqi(1)
=
dbqidz
∣∣∣∣z=1
= 1−1
bpi(1)
= limz→1
1
(1− z)bpi(z)
![Page 70: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/70.jpg)
RECURRENTi ∈ Ω Ri = 1⇔
POSITIVE RECURRENT ⇔ τi < ∞
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
⇔ bpi(1) = ∞
⇔ limz→1
(1− z)bpi(z) > 0
= bqi(1)
=
dbqidz
∣∣∣∣z=1
= 1−1
bpi(1)
= limz→1
1
(1− z)bpi(z)
![Page 71: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/71.jpg)
RECURRENCE & SPECTRUM
![Page 72: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/72.jpg)
RECURRENCE & SPECTRUM
=
X
n≥0
Pn
iizn
P bpi(z)
![Page 73: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/73.jpg)
RECURRENCE & SPECTRUM
=
X
n≥0
Pn
iizn
P bpi(z)
= (1− zP )−1
ii
![Page 74: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/74.jpg)
RECURRENCE & SPECTRUM
P bpi(z) = (1− zP )−1
ii
![Page 75: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/75.jpg)
RECURRENCE & SPECTRUM
P bpi(z)
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii
![Page 76: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/76.jpg)
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?τi = lim
z→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii
![Page 77: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/77.jpg)
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?τi = lim
z→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 78: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/78.jpg)
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 79: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/79.jpg)
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 80: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/80.jpg)
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 81: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/81.jpg)
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 82: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/82.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 83: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/83.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 84: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/84.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
RECURRENT⇔⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 85: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/85.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN IN
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
RECURRENT⇔⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 86: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/86.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN IN
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 87: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/87.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN IN
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
![Page 88: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/88.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN INP FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Under quite general conditions P becomes self-adjoint with kPk 1
![Page 89: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/89.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN INP FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Recurrence ONLY depends on the spectral decomposition around λ = 1
Under quite general conditions P becomes self-adjoint with kPk 1
![Page 90: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/90.jpg)
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN INP FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Recurrence ONLY depends on the spectral decomposition around λ = 1
FINITE systems: RECURRENT POSITIVE RECURRENT⇒
Under quite general conditions P becomes self-adjoint with kPk 1
![Page 91: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/91.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
![Page 92: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/92.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
![Page 93: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/93.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
![Page 94: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/94.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
![Page 95: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/95.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
source: http://physik.uni-paderborn.de/?id=178571
![Page 96: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/96.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
RW
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
source: http://physik.uni-paderborn.de/?id=178571
![Page 97: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/97.jpg)
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
RW QW
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
source: http://physik.uni-paderborn.de/?id=178571
![Page 98: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/98.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
![Page 99: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/99.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Simple models for quantum dynamics
![Page 100: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/100.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Simple models for quantum dynamics
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
![Page 101: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/101.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
![Page 102: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/102.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
![Page 103: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/103.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
![Page 104: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/104.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
![Page 105: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/105.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
source: http://physik.uni-paderborn.de/?id=178571
![Page 106: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/106.jpg)
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
source: http://physik.uni-paderborn.de/?id=178571
![Page 107: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/107.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
![Page 108: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/108.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 109: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/109.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 110: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/110.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 111: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/111.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 112: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/112.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 113: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/113.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 114: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/114.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 115: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/115.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
![Page 116: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/116.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
![Page 117: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/117.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
![Page 118: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/118.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
FIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
![Page 119: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/119.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
FIRST RETURN PROB.
in STEPSnFIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
![Page 120: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/120.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
PROBLEM: the quantum measurement to check the return after every step collapses the state altering irreversibly the “natural” evolution
FIRST RETURN PROB.
in STEPSnFIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
![Page 121: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/121.jpg)
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
PROBLEM: the quantum measurement to check the return after every step collapses the state altering irreversibly the “natural” evolution
We will take the collapse as an intrinsic ingredient of monitored quantum recurrence.
This is in QM spirit, which gives to measurements a role absent in classical physics
FIRST RETURN PROB.
in STEPSnFIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
![Page 122: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/122.jpg)
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ|
![Page 123: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/123.jpg)
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ|
Qφψ
![Page 124: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/124.jpg)
PROBABILITY OF FINDING φ
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Qφψ
![Page 125: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/125.jpg)
PROBABILITY OF FINDING φ
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Two possible results when measuring at state φ ψ
Qφψ
![Page 126: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/126.jpg)
PROBABILITY OF FINDING φ
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Two possible results when measuring at state φ ψ
φ is found:Pφψ
k · k= φψ
COLLAPSE−−−−−−−−→
Qφψ
![Page 127: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/127.jpg)
PROBABILITY OF FINDING φ
Qφψ
kQφψk
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Two possible results when measuring at state φ ψ
φ is found:Pφψ
k · k= φψ
COLLAPSE−−−−−−−−→
φ is NOT found:Qφψ
k · kψ
COLLAPSE−−−−−−−−→
Qφψ
![Page 128: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/128.jpg)
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
![Page 129: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/129.jpg)
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
Dynamics perturbed by measurements≡
![Page 130: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/130.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
Dynamics perturbed by measurements≡
![Page 131: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/131.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
Dynamics perturbed by measurements≡
![Page 132: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/132.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
![Page 133: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/133.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
![Page 134: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/134.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
![Page 135: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/135.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
![Page 136: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/136.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
NO ψ NO ψ
step 3−−−−−→ · · ·
step n−1−−−−−−−→ eUn−1ψ
![Page 137: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/137.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
step n
−−−−−→ U eUn−1ψNO ψ NO ψ
step 3−−−−−→ · · ·
step n−1−−−−−−−→ eUn−1ψ
![Page 138: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/138.jpg)
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
FIRST RETURN
PROBABILITY
in STEPSn
= |hψ|U eUn−1ψi|2FIRST TIME
Prob(ψn steps−−−−−→ ψ)
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
step n
−−−−−→ U eUn−1ψNO ψ NO ψ
step 3−−−−−→ · · ·
step n−1−−−−−−−→ eUn−1ψ
![Page 139: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/139.jpg)
QW RECURRENCE
![Page 140: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/140.jpg)
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
![Page 141: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/141.jpg)
projection onto ψ⊥
eU = QψU
FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
FIRST TIMEProb(ψ
n steps−−−−−→ ψ)= |a
(n)ψ |2
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
![Page 142: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/142.jpg)
projection onto ψ⊥
eU = QψU
FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
FIRST TIMEProb(ψ
n steps−−−−−→ ψ)= |a
(n)ψ |2
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
RETURNPROBABILITY
Rψ = Prob(ψ → ψ) =X
n≥1
|a(n)ψ |2
EXPECTED RETURN TIME
τψ =
X
n≥1
n |a(n)ψ |2
![Page 143: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/143.jpg)
projection onto ψ⊥
eU = QψU
FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
FIRST TIMEProb(ψ
n steps−−−−−→ ψ)= |a
(n)ψ |2
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
is RECURRENT if ψ Rψ = 1
POSITIVE RECURRENT if τψ < ∞
RETURNPROBABILITY
Rψ = Prob(ψ → ψ) =X
n≥1
|a(n)ψ |2
EXPECTED RETURN TIME
τψ =
X
n≥1
n |a(n)ψ |2
![Page 144: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/144.jpg)
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
![Page 145: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/145.jpg)
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
|a|2 + |c|2 = 1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
![Page 146: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/146.jpg)
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
|a|2 + |c|2 = 1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
UNBIASED RW
CRITICAL DIMENSION
D= 3
![Page 147: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/147.jpg)
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
|a|2 + |c|2 = 1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
UNBIASED COINED QW
|a|2 = |b|2 = |c|2 = |d|2 =1
2
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
UNBIASED RW
CRITICAL DIMENSION
D= 3
Every state is TRANSIENT
already for !!! D= 1
![Page 148: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/148.jpg)
Example: cyclic shift
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
![Page 149: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/149.jpg)
Example: cyclic shift
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
![Page 150: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/150.jpg)
Example: cyclic shift
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
Prob(ψ2 steps−−−−−→ ψ) =
9
16FIRST TIMEa(2)ψ = hψ|U eUψi = −
3
4
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
![Page 151: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/151.jpg)
Example: cyclic shift
<
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
Prob(ψ2 steps−−−−−→ ψ) =
9
16FIRST TIMEa(2)ψ = hψ|U eUψi = −
3
4
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
![Page 152: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/152.jpg)
Example: cyclic shift
QUANTUM PARADOX
FIRST return probabilities can be greater than return probabilities!!!
<
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
Prob(ψ2 steps−−−−−→ ψ) =
9
16FIRST TIMEa(2)ψ = hψ|U eUψi = −
3
4
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
![Page 153: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/153.jpg)
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
![Page 154: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/154.jpg)
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
![Page 155: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/155.jpg)
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
![Page 156: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/156.jpg)
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
For amplitudes
instead of
probabilities!!!
![Page 157: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/157.jpg)
=
X
n≥1
|a(n)ψ |2Rψ
=
X
n≥1
n|a(n)ψ |2
RETURNPROBABILITY
EXPECTED RETURN TIME τψ
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
For amplitudes
instead of
probabilities!!!
![Page 158: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/158.jpg)
=
X
n≥1
|a(n)ψ |2Rψ
=
X
n≥1
n|a(n)ψ |2
RETURNPROBABILITY
EXPECTED RETURN TIME τψ
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
=
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πi
For amplitudes
instead of
probabilities!!!
![Page 159: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/159.jpg)
RECURRENCE & SPECTRUM
![Page 160: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/160.jpg)
RECURRENCE & SPECTRUM
U bµψ(z) =X
n≥0
hψ|Unψi zn
![Page 161: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/161.jpg)
RECURRENCE & SPECTRUM
U bµψ(z) =X
n≥0
hψ|Unψi zn
= hψ|(1− zU)−1ψi
![Page 162: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/162.jpg)
RECURRENCE & SPECTRUM
U bµψ(z) = hψ|(1− zU)−1ψi
![Page 163: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/163.jpg)
RECURRENCE & SPECTRUM
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
![Page 164: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/164.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
![Page 165: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/165.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
U unitary ⇒spectrum in
unit circle
eiθ
![Page 166: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/166.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
![Page 167: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/167.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+
Zdψ(eiθ)
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 168: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/168.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 169: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/169.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 170: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/170.jpg)
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 171: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/171.jpg)
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 172: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/172.jpg)
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 173: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/173.jpg)
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
![Page 174: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/174.jpg)
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS
dimH = ∞
![Page 175: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/175.jpg)
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS
τψ = number of EIGENVECTORS
dimH = ∞
![Page 176: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/176.jpg)
RECURRENCE & SPECTRUM
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
![Page 177: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/177.jpg)
RECURRENCE & SPECTRUM
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
Recurrence depends an ALL the spectral decomposition
![Page 178: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/178.jpg)
RECURRENCE & SPECTRUM
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
![Page 179: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/179.jpg)
RECURRENCE & SPECTRUM
QUANTIZATION of EXPECTED RETURN TIME!!!
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
![Page 180: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/180.jpg)
RECURRENCE & SPECTRUM
QUANTIZATION of EXPECTED RETURN TIME!!!
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
θα
eiθ e
iα
baψ(eiθ)
=∆α
2πτψ
WINDING NUMBER of baψ(eiθ)
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
![Page 181: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/181.jpg)
EXPECTED RETURN TIME: Topological meaning INTEGER
RECURRENCE & SPECTRUM
QUANTIZATION of EXPECTED RETURN TIME!!!
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
θα
eiθ e
iα
baψ(eiθ)
=∆α
2πτψ
WINDING NUMBER of baψ(eiθ)
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
![Page 182: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/182.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
![Page 183: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/183.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
-1 10
-1 10
Example: site recurrence in D coined QW1
SUBESPACE RECURRENCE
![Page 184: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/184.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
-1 10
-1 10
Example: site recurrence in D coined QW1
|0i |"iψ =
−→
Prob???
span|0i |"i, |0i |#iV =
SUBESPACE RECURRENCE
![Page 185: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/185.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
-1 10
-1 10
Example: site recurrence in D coined QW1
|0i |"iψ =
−→
Prob???
span|0i |"i, |0i |#iV =
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
![Page 186: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/186.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
The generating functions become matrix functions acting on V
![Page 187: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/187.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
The generating functions become matrix functions acting on V
![Page 188: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/188.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ
The generating functions become matrix functions acting on V
![Page 189: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/189.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ
Rψ(V ) = Prob(ψ → V ) -RETURNPROBABILITYV
The generating functions become matrix functions acting on V
![Page 190: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/190.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ=
Z2π
0
k bψ(θ)k2 dθ
2πRψ(V ) = Prob(ψ → V ) -RETURN
PROBABILITYV
The generating functions become matrix functions acting on V
![Page 191: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/191.jpg)
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ=
Z2π
0
k bψ(θ)k2 dθ
2πRψ(V ) = Prob(ψ → V ) -RETURN
PROBABILITYV
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi EXPECTED -RETURN TIMEV
The generating functions become matrix functions acting on V
![Page 192: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/192.jpg)
1/ dimV
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
![Page 193: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/193.jpg)
1/ dimV
V
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
![Page 194: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/194.jpg)
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
![Page 195: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/195.jpg)
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
Averaging over we find again “topological integers”ψ ∈ Vτψ(V )
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
![Page 196: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/196.jpg)
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
Averaging over we find again “topological integers”ψ ∈ Vτψ(V )
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
hτψ(V )iψ∈V =N
dimV
N =
WINDING NUMBER
of detbaV (eiθ)
θ
eiθ
α
eiα
detbaV (eiθ)
![Page 197: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/197.jpg)
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
Averaging over we find again “topological integers”ψ ∈ Vτψ(V )
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
hτψ(V )iψ∈V =N
dimV
N =
WINDING NUMBER
of detbaV (eiθ)
θ
eiθ
α
eiα
detbaV (eiθ)
QUANTIZATION of MEAN EXPECTED -RETURN TIME!!!
Topological meaning INTEGER MULTIPLE of
V
1/ dimV
![Page 198: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/198.jpg)
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
Example: site recurrence in D coined QW1
![Page 199: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/199.jpg)
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
|a|2 + |c|2 = 1
R|xi|si(V ) =2
π|c|2
|ac|+ (1− 2|a|2) arcsin |a|
SITE RECURRENCE
Example: site recurrence in D coined QW1
![Page 200: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/200.jpg)
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
STATE RECURRENCE
|a|2 + |c|2 = 1
R|xi|si(V ) =2
π|c|2
|ac|+ (1− 2|a|2) arcsin |a|
SITE RECURRENCE
Example: site recurrence in D coined QW1
![Page 201: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/201.jpg)
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
STATE RECURRENCE
|a|2 + |c|2 = 1
R|xi|si(V ) =2
π|c|2
|ac|+ (1− 2|a|2) arcsin |a|
SITE RECURRENCE
Example: site recurrence in D coined QW1
As intuition suggests, . Is this a general fact?Prob(ψ → V ) ≥ Prob(ψ → ψ)
![Page 202: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/202.jpg)
Example: combined shifts
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
![Page 203: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/203.jpg)
Example: combined shifts
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
ψ = α|0i+ |"ip
2+ β|#iFor and we obtainV = span|0i+ |"i, |#i
Rψ = Prob(ψ → ψ) =1− 1
2|α|2
1 + 1
2|α|2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
|α|
Rψ(V ) = Prob(ψ → V ) = 3
4− 1
4|α|2
![Page 204: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/204.jpg)
Example: combined shifts
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
ψ = α|0i+ |"ip
2+ β|#iFor and we obtainV = span|0i+ |"i, |#i
Rψ = Prob(ψ → ψ) =1− 1
2|α|2
1 + 1
2|α|2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
|α|
Rψ(V ) = Prob(ψ → V ) = 3
4− 1
4|α|2
![Page 205: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/205.jpg)
Example: combined shifts
QUANTUM PARADOX
Return probability to a subspace can be smaller than to the state!!!
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
ψ = α|0i+ |"ip
2+ β|#iFor and we obtainV = span|0i+ |"i, |#i
Rψ = Prob(ψ → ψ) =1− 1
2|α|2
1 + 1
2|α|2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
|α|
Rψ(V ) = Prob(ψ → V ) = 3
4− 1
4|α|2
![Page 206: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/206.jpg)
RW vs QW
Random Walks Quantum Walks
Spectral shortcut NOT always applicable
Stochastic Self-adjoint Spectrum on [-1,1]
Spectral shortcut always applicable
Unitary Spectrum on unit circle
ONLY spectrum around 1 matters for recurrence ALL the spectrum matters for recurrence
Recurrence Singular part Recurrence ONLY singular part
eigenvector Positive recurrence
with eigenvalue 1
ONLY finite eigenvectors Positive recurrence
with ANY eigenvalue
Finite system ONLY recurrent states ONLY positive
Finite-dim system recurrent states
Expected return time is NOT quantized Expected return time is quantized
First return probabilities
are NOT greater than return probabilities
First return probabilities
can be greater than return probabilities
Return probability to a subset
is NOT smaller than to the initial state
Return probability to a subspace
can be smaller than to the initial state
⇔;:
⇔⇔
; ⇒
? ⇒ ⇒
∃
![Page 207: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/207.jpg)
RW vs QW
Random Walks Quantum Walks
Spectral shortcut NOT always applicable
Stochastic Self-adjoint Spectrum on [-1,1]
Spectral shortcut always applicable
Unitary Spectrum on unit circle
ONLY spectrum around 1 matters for recurrence ALL the spectrum matters for recurrence
Recurrence Singular part Recurrence ONLY singular part
eigenvector Positive recurrence
with eigenvalue 1
ONLY finite eigenvectors Positive recurrence
with ANY eigenvalue
Finite system ONLY recurrent states ONLY positive
Finite-dim system recurrent states
Expected return time is NOT quantized Expected return time is quantized
First return probabilities
are NOT greater than return probabilities
First return probabilities
can be greater than return probabilities
Return probability to a subset
is NOT smaller than to the initial state
Return probability to a subspace
can be smaller than to the initial state
⇔;:
⇔⇔
; ⇒
? ⇒ ⇒
∃
![Page 208: RECURRENCE: RANDOM WALKS vs QUANTUM WALKS€¦ · George Pólya George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park He reduced the](https://reader034.fdocuments.us/reader034/viewer/2022051806/5ffe75c59c890d40395c34b8/html5/thumbnails/208.jpg)
RECURRENCE COLLABORATORS
Reinhard WernerLeibniz U Hannover
Albert WernerFreie U Berlin
Alberto GrünbaumUC Berkeley
Jon WilkeningUC Berkeley
Jean BourgainIAS Princeton